WASAPI generally provides lower and more consistent latency recording than DirectSound, so I would recommend using that whenever possible (it is the default unless disabled via BASS_CONFIG_REC_WASAPI).
To get a recording latency value, you could try measuring how long it takes for data to start arriving from a recording. For example, something like this:
HRECORD record = BASS_RecordStart(freq, chans, 0, NULL, NULL); // create recording channel (no RECORDPROC)
DWORD start = timeGetTime(), avail;
while (!(avail = BASS_ChannelGetData(record, NULL, BASS_DATA_AVAILABLE))) Sleep(1); // wait for data to become available
double latency = (timeGetTime() - start) / 1000.0 - BASS_ChannelBytes2Seconds(record, avail); // calculate latency in seconds
BASS_ChannelFree(record); // free the recording